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NumberTheory

  

CarmichaelLambda

  

Carmichael's lambda function

 

Calling Sequence

Parameters

Description

Examples

Compatibility

Calling Sequence

CarmichaelLambda(n)

 

lambda(n)

λn

Parameters

n

-

positive integer

Description

• 

The size of the largest cyclic group generated by gimodn is given by CarmichaelLambda(n).

• 

Alternatively, CarmichaelLambda(n) is the smallest integer i such that for all g coprime to n, gi is congruent to 1 modulo n.

• 

lambda is an alias for CarmichaelLambda.

• 

You can enter the command lambda using either the 1-D or 2-D calling sequence. For example, lambda(8) is equivalent to λ8.

Examples

withNumberTheory:

seqTotienti,i=1..7

1,1,2,2,4,2,6

(1)

seqCarmichaelLambdai,i=1..7

1,1,2,2,4,2,6

(2)

CarmichaelLambda8

2

(3)

Totient8

4

(4)

λ21

6

(5)

Totient21

12

(6)

CarmichaelLambdak

CarmichaelLambdak

(7)

Carmichael's theorem states that gλn is congruent to 1 modulo n if g and n are coprime.

dCarmichaelLambda112

d12

(8)

seq`if`igcdg,112=1,gdmod112,NULL,g=1..111

1

(9)

Compatibility

• 

The NumberTheory[CarmichaelLambda] command was introduced in Maple 2016.

• 

For more information on Maple 2016 changes, see Updates in Maple 2016.

See Also

NumberTheory

NumberTheory[Totient]